Finite Group DatabasebetaFinite Group Databasebeta| order | 1152 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C9 · Schur multiplier (H2) = C25 |
| direct product | C9×#2972 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C9 | C24 |
| commutator | C2 | C23×C8×C9 |
| Frattini, cosocle | C2×C3×C4 | C24×C3 |
| socle | C22×C3 | C23×C3×C4 |
| prime radical | C22×C3×D4 | C3×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C9 ◁ G | |
750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 288 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4,5,6,7,8,9)(10,11)(12,16)(13,15)(14,17)(18,19,20,22,21,23,24,25);(10,12)(11,14)(13,17)(15,16);(12,17)(14,16);(11,15)(14,16)612 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1152, 1704);